Arrow’s theorem in judgment aggregation
Franz Dietrich
(University of Maastricht)
and
Christian List
(LSE)
Political Economy and Public Policy Series The Suntory Centre
Suntory and Toyota International Centres for Economics and Related Disciplines
London School of Economics and Political Science Houghton Street
London WC2A 2AE PEPP/13
October 2005 Tel: (020) 7955 6674
© The author. All rights reserved. Short sections of text, not to exceed two paragraphs, may be quoted without explicit permission provided that full credit, including © notice, is given to the source
Arrow’s theorem in judgment aggregation
Franz Dietrich and Christian List14/2005, revised 9/2005
In response to recent work on the aggregation of individual judgments on logically connected propositions into collective judgments, it is often asked whether judgment aggregation is a special case of Arrowian preference aggregation. We argue the op-posite. After proving a general impossibility result on judgment aggregation, we construct an embedding of preference aggregation into judgment aggregation and prove Arrow’s theorem as a corollary of our result. Although we provide a new proof of Arrow’s theorem, our main aim is to identify the analogue of Arrow’s theorem in judgment aggregation, to clarify the relation between judgment and preference aggregation and to illustrate the generality of the judgment aggregation model. JEL Classi…cation: D70, D71
1
Introduction
A new aggregation problem has recently received much attention. How can the judgments of several individuals on logically connected propositions be ag-gregated into corresponding collective judgments? To illustrate the di¢ culties involved in judgment aggregation, suppose a three-member committee has to make collective judgments on three connected propositions:
a: "Carbon dioxide emissions are above the threshold x."
a ! b: "If carbon dioxide emissions are above the threshold x, then there will be global warming."
b: "There will be global warming."
a a !b b
Individual 1 True True True
Individual 2 True False False Individual 3 False True False
Majority True True False
Table 1: The discursive dilemma
As shown in Table 1, the …rst committee member accepts all three proposi-tions; the second acceptsabut rejectsa!b andb; the third acceptsa!b but rejectsaandb. Then the judgments of each committee member are individually consistent, and yet the majority judgments on the propositions are inconsistent: a majority acceptsa, a majority acceptsa!b, but a majority rejectsb.
1We thank Richard Bradley, Ruvin Gekker, Ron Holzman and Philippe Mongin for
com-ments and suggestions. Addresses: F. Dietrich, Department of Quantitative Economics, Uni-versity of Maastricht, P.O. Box 616, 6200 MD Maastricht, The Netherlands; C. List, Dep-tartment of Government, London School of Economics, London WC2A 2AE, U.K.
This problem –the so-calleddiscursive dilemma (Pettit 2001) –has led to a growing literature on the possibility of consistent judgment aggregation under various conditions. List and Pettit (2002, 2004) have developed a …rst model of judgment aggregation based on propositional logic and proved an impossibility result, followed by several stronger impossibility results (Pauly and van Hees 2004; Dietrich 2004a; Gärdenfors 2004; van Hees 2004; Nehring and Puppe 2005; Dietrich and List 2005; Dokow and Holzman 2005) and possibility results (Bovens and Rabinowicz 2004; Dietrich 2004a; List 2003, 2004, 2005; Pigozzi 2004). Generalizing the propositional logic framework, Dietrich (2004b) has developed a model of judgment aggregation in general logics, also used in the present paper, which allows the representation of a larger class of aggregation problems.
Although there are obvious di¤erences between judgment aggregation and the more familiar problem of preference aggregation, the recent results resemble earlier results in social choice theory. The discursive dilemma resembles Con-dorcet’s paradox of cyclical majority preferences, and the various impossibility theorems on judgment aggregation resemble Arrow’s theorem on preference ag-gregation. This has led critics to ask whether the recent work on judgment aggregation is a reinvention of the wheel.
In response, it can be argued that the work on judgment aggregation is not a reinvention, but a generalization. The preference aggregation model of Condorcet and Arrow can be embedded into a judgment aggregation model by representing preference orderings as sets of binary ranking judgments (List and Pettit 2004; List 2003, fn. 4), whereas the converse embedding cannot easily be achieved. (This embedding claim applies only to the ordinal preference-relation-based strand of Arrowian social choice theory, but not to the cardinal welfare-function-based strand.)
In this paper, we reinforce this argument. After introducing the judgment aggregation model in general logics and proving a general impossibility result (stated in terms of two theorems), we construct an explicit embedding of pref-erence aggregation into judgment aggregation and prove Arrow’s theorem (for strict preferences) as a direct corollary of our result on judgment aggregation. Our aim is not primarily to provide a new proof of Arrow’s theorem, but to identify the analogue of Arrow’s theorem in judgment aggregation, to clarify the logical relation between judgment and preference aggregation and to high-light the logical structure underlying Arrow’s result.
Related results were given by List and Pettit (2004), who derived a sim-ple impossibility theorem on preference aggregation from an earlier result on judgment aggregation, and Nehring (2003), who proved an Arrow-like theorem in the related framework of property spaces. But neither result fully matches Arrow’s theorem. List and Pettit’s result requires additional neutrality and anonymity conditions, but no Pareto principle. Nehring’s result requires an additional monotonicity condition.
2
The judgment aggregation model
We consider a group of individuals 1;2; : : : ; n (n 2). The group has to make collective judgments on logically connected propositions.
Formal logic. Propositions are represented in an appropriate formal logic. A logic (with negation symbol :) is a pair (L; ) such that
L is a non-empty set ofpropositions, where p2L implies:p2L, is an entailment relation ( P(L) L), where, for each set of propositions A L and each proposition p2L,A pis read as "A entails p" (we write p q as an abbreviaton for fpg q).
A setA Lisinconsistent if A pand A :pfor somep2L, andconsistent
otherwise. A set A L is minimal inconsistent if it is inconsistent and every proper subset B ( A is consistent. A proposition p 2 L is contingent if fpg
and f:pg are consistent. We require our logic to satisfy the following minimal conditions:
(L1) For all p2L, p p.
(L2) For all p2L and A B L, if A p then B p.
(L3) ; is consistent, and each consistent set A L has a consistent superset B L containing a member of each pair p;:p2L.
Many di¤erent logics satisfy conditions L1 to L3, including standard proposi-tional logic, standard modal and condiproposi-tional logics and, for the present purposes, predicate logic, as de…ned below.
The agenda. The agenda is a non-empty subset X L, interpreted as the set of propositions on which judgments are to be made, where X is a union of proposition-negation pairs fp;:pg (with p not itself a negated proposition). For simplicity, we assume that double negations cancel each other out, i.e. ::p
stands for p.2 In the example above, the agenda is X = fa;:a; b;:b; a ! b;:(a!b)gin standard propositional logic (or alternatively in a simple condi-tional logic). We consider agendas X with di¤erent types of interconnections. For anyp; q 2X, we write p qif fp;:qg [Y is inconsistent for someY X
consistent withp and with :q.3
X isminimally connected if
(i) there exists a minimal inconsistent set Y X with jYj 3, and (ii) there exists a minimal inconsistent set Y X such that(YnZ)[
f:z :z 2Zg is consistent for some subset Z Y of even size.
X ispath-connected if, for every contingent p; q 2X, there exist p1;
p2; :::; pk2X (with p=p1 and q =pk) such thatp1 p2, p2 p3, ...,
pk 1 pk.
2When we use the negation symbol
: hereafter, we mean a modi…ed negation symbol , where p:=:pifpis unnegated and p:=qifp=:q for someq.
3For non-paraconsistent logics (in the sense of L4 in Dietrich 2004b),
fp;:qg [Y is incon-sistent if and only iffpg [Y q.
Dokow and Holzman (2005) have recently introduced an algebraic condition on …nite agendas, which is equivalent to part (ii) of minimal connectedness, as Ron Holzman has indicated to us. If the logic is compact, path-connectedness is equivalent to Nehring and Puppe’s (2005) total blockedness; in the general case, path-connectedness is weaker.
The agenda of our example above is minimally connected, but not path-connected. Below we show that preference aggregation problems can be repre-sented by agendas that are both minimally connected and path-connected.
Individual judgment sets. Each individual i’sjudgment set is a subset Ai
X, wherep2Ai means that individualiaccepts propositionp. A judgment set
Ai is consistent if it is a consistent set as de…ned above; Ai is complete if, for every proposition p2 X, p2Ai or :p2Ai. Apro…le (of individual judgment
sets)is an n-tuple (A1; : : : ; An).
Aggregation rules. A (judgment) aggregation rule is a function F that as-signs to each admissible pro…le (A1; : : : ; An) a single collective judgment set
F(A1; : : : ; An) = A X, where p 2 A means that the group accepts propo-sition p. The set of admissible pro…les is called the domain of F, denoted
Domain(F). Examples of aggregation rules are the following.
Propositionwise majority voting. For each (A1; :::; An), F(A1; :::; An)
=fp2X : more individualsi havep2Ai than p =2Aig.
Dictatorship of individual i:For each (A1; :::; An), F(A1; :::; An) =Ai.
Inverse dictatorship of individual i: For each (A1; :::; An),F(A1; :::; An)
=f:p:p2Aig.
Regularity conditions on aggregation rules. We impose the following condi-tions on the inputs and outputs of aggregation rules.
Universal domain. The domain of F is the set of all possible pro…les of
complete and consistent individual judgment sets.
Collective rationality. F generates complete and consistent collective judg-ment sets.
Propositionwise majority voting, dictatorships and inverse dictatorships sat-isfy universal domain, but only dictatorships generally satsat-isfy collective ratio-nality. As the discursive dilemma example of Table 1 shows, propositionwise majority voting sometimes generates inconsistent collective judgment sets. In-verse dictatorships satisfy collective rationality only in special cases (i.e. when the agenda is symmetrical: for every consistent Z X, f:p : p 2 Zg is also consistent).
3
An impossibility result on judgment
aggre-gation
Are there any non-dictatorial judgment aggregation rules satisfying universal domain and collective rationality? The following conditions are frequently used in the literature.
Independence. For any propositionp2X and pro…les (A1; : : : ; An);(A1; : : : ;
An) 2 Domain(F), if [for all individuals i, p 2 Ai if and only if p 2 Ai] then [p2F(A1; : : : ; An) if and only ifp 2F(A1; : : : ; An)].
Systematicity. For any propositions p; q 2 X and pro…les (A1; : : : ; An);
(A1; : : : ; An) 2 Domain(F), if [for all individuals i, p 2 Ai if and only if
q2Ai] then [p2F(A1; : : : ; An) if and only if q 2F(A1; : : : ; An)].
Unanimity principle. For any pro…le (A1; : : : ; An) 2 Domain(F) and any propositionp2X, if p2Ai for all individuals i, thenp2F(A1; : : : ; An).
Independence requires that the collective judgment on each proposition should depend only on individual judgments on that proposition. Systematic-ity strengthens independence by requiring in addition that the same pattern of dependence should hold for all propositions (a neutrality condition). The una-nimity principle requires that if all individuals accept a proposition then this proposition should also be collectively accepted. The following result holds.
Proposition 1. For a minimally connected agenda X, an aggregation ruleF
satis…es universal domain, collective rationality, systematicity and the unanim-ity principle if and only if it is a dictatorship of some individual.
Proof. All proofs are given in the appendix.
Proposition 1 is related to an earlier result by Dietrich (2004b), which re-quires an additional assumption on the agenda X but no unanimity principle (the additional assumption is thatXis alsoasymmetrical: for some inconsistent
Z X, f:p :p2Zg is consistent). This result, in turn, generalizes an earlier result on systematicity by Pauly and van Hees (2004).
From Proposition 1, we can derive two new results of interest. The …rst is a generalization of List and Pettit’s (2002) theorem on the non-existence of an ag-gregation rule satisfying universal domain, collective rationality, systematicity and anonymity (i.e. invariance of the collective judgment set under permu-tations of the given pro…le of individual judgment sets). Our result extends the earlier impossibility result to any minimally connected agenda and weakens anonymity to the requirement that there is no dictator or inverse dictator.
Theorem 1. For a minimally connected agenda X, an aggregation rule F
satis…es universal domain, collective rationality and systematicity if and only if it is a (possibly inverse) dictatorship of some individual.
Moreover, the agenda assumption of Theorem 1 is maximally weak ifn 3
and the agenda is …nite or the logic is compact (andX contains at least one con-tingent proposition), i.e. minimal connectedness is necessary for characterizing (possibly inverse) dictatorships by the conditions of Theorem 2.4
The second result we can derive from Proposition 1 is the analogue of Arrow’s theorem in judgment aggregation, from which we subsequently derive Arrow’s theorem on preference aggregation as a corollary. Note the following lemma.
Lemma 1. For a path-connected agenda X, an aggregation rule F
satisfy-ing universal domain, collective rationality, independence and the unanimity principle also satis…es systematicity.
Let us call an agenda strongly connected if it is both minimally connected and path-connected. Using Lemma 1, Proposition 1 now implies the following impossibility result.
Theorem 2. For a strongly connected agenda X, an aggregation rule F
sat-is…es universal domain, collective rationality, independence and the unanimity principle if and only if it is a dictatorship of some individual.
Dokow and Holzman (2005) have recently shown that, if n 3 and the agenda is …nite and contains only contingent propositions, the agenda assump-tion of Theorem 2 is maximally weak, i.e. strong connectedness is necessary for characterizing dictatorships by the conditions of Theorem 2. In fact, the necessity holds whenevern 3and the agenda is …nite or the logic is compact (and X contains at least one contingent proposition). A related result with an additional monotonicity condition has been proved by Nehring and Puppe (2005).
Our impossibility results continue to hold under slightly generalized de…ni-tions of minimally connected and strongly connected agendas.5
Of course, it is debatable whether and when independence or systematicity are plausible requirements on judgment aggregation. The literature contains extensive discussions of these conditions and their possible relaxations. In our
4It can then be shown that, if X is not minimally connected, there exists an aggregation
rule that satis…es universal domain, collective rationality and systematicity and is not a (possibly inverse) dictatorship. LetM be a subset off1; :::; ng of odd size at least 3. If part (i) of minimal connectedness is violated, then majority voting among the individuals inM
satis…es all requirements. If part (ii) is violated, the aggregation ruleF with universal domain de…ned byF(A1; :::; An) :=fp2 X : the number of individualsi 2M withp2Ai is oddg
satis…es all requirements.
5In the de…nition of minimal connectedness, (i) can be weakened to the following: (i*) there
is an inconsistent setY X with pairwise disjoint subsetsZ1; Z2; Z3such that(YnZj)[f:p: p2Zjgis consistent for anyj2 f1;2;3g. In the de…nition of strong connectedness (by (i), (ii)
and path-connectedness), (i) can be dropped altogether, as path-connectedness implies (i*). In the de…nitions of minimal connectedness and strong connectedness, (ii) can be weakened to ( *) in Dietrich (2004b).
view, the importance of Theorems 1 and 2 lies not so much in establishing the impossibility of consistent judgment aggregation, but rather in indicating what conditions must be relaxed in order to make consistent judgment aggregation possible. The theorems describe boundaries of the logical space of possibilities.
4
Arrow’s theorem
We now show that Arrow’s theorem (for strict preferences) can be stated in the judgment aggregation model, where it is a direct corollary of Theorem 2. We consider a standard Arrowian preference aggregation model, where each indi-vidual has a strict preference ordering (asymmtrical, transitive and connected, as de…ned below) over a set of options K =fx; y; z; :::g with jKj 3. We em-bed this model into our judgment aggregation model by representing preference orderings as sets of binary ranking judgments in a simple predicate logic.6
A simple predicate logic for representing preferences. We consider a predi-cate logic with constants x; y; z; :::2K (representing the options), variablesv,
w, v1, v2 , ..., identity symbol =, a two-place predicate P (representing strict
preference), logical connectives : (not), ^ (and), _ (or), ! (if-then), and uni-versal quanti…er8. Formally, L is the smallest set such that
L contains all propositions of the forms P and = , where and are constants or variables, and
whenever L contains two propositions p and q, then L also contains
:p, (p^q), (p_q),(p!q) and (8v)p, wherev is any variable.
Notationally, we drop brackets when there is no ambiguity. The entailment relation is de…ned as follows. For any set A L and any propositionp2L,
A pif and only if A[Z entails pin the standard sense of predicate logic,
whereZ is the set of rationality conditions on strict preferences:
(8v1)(8v2)(v1P v2 ! :v2P v1) ("asymmetry");
(8v1)(8v2)(8v3)((v1P v2^v2P v3)!v1P v3) ("transitivity");
(8v1)(8v2)(:v1=v2 !(v1P v2_v2P v1)) ("connectedness").7
The agenda. The preference agenda is the set X of all propositions of the formsxP y;:xP y 2L, wherex and y are distinct constants.
Lemma 2. The preference agenda X is strongly connected.
6Although we consider strict preferences for simplicity, a similar embedding is also possible
for weak preferences.
7For technical reasons,Zalso contains, for each pair of distinct constantsx; y, the condition
The correspondence between preference orderings and judgment sets. It is easy to see that each (asymmetrical, transitive and connected) preference order-ing over K can be represented by a unique complete and consistent judgment set in X and vice-versa, where individual i strictly prefers xto y if and only if
xP y2Ai. For example, if individualistrictly prefersxtoytoz, this is uniquely represented by the judgment setAi =fxP y; yP z; xP z;:yP x;:zP y;:zP xg.
The correspondence between Arrow’s conditions and conditions on judgment aggregation. For the preference agenda, the conditions of universal domain, collective rationality, independence and the unanimity principle ("Pareto"), as stated above, exactly match the standard conditions of Arrow’s theorem.
As the preference agenda is strongly connected, Arrow’s theorem now follows from Theorem 2.
Corollary 1. (Arrow’s theorem) For the preference agenda X, an aggregation rule F satis…es universal domain, collective rationality, independence and the unanimity principle if and only if it is a dictatorship of some individual.
5
Concluding remarks
After proving a general impossibility result on judgment aggregation, stated in terms of Theorems 1 and 2, we have shown that Arrow’s theorem (for strict preferences) is a corollary of this result, speci…cally of Theorem 2 applied to the aggregation of binary ranking judgments in a simple predicate logic.
This …nding illustrates the generality of the judgment aggregation model. The model can represent a large class of judgment aggregation problems in general logics –all logics satisfying conditions L1 to L3 –of which a predicate logic for representing preferences is a special case. Other logics to which the model applies are propositional, modal or conditional logics and predicate logics representing relational structures other than preference orderings. Impossibility and possibility results on judgment aggregation, such as Theorems 1 and 2, can apply to aggregation problems in all these logics.
An alternative, very general model of aggregation is the one introduced by Wilson (1975) and recently used by Dokow and Holzman (2005), where a group has to determine its yes/no views on several issues based on the group members’ views on these issues (subject to feasibility constraints). Wilson’s model can also be represented in our model; here Dokow and Holzman’s results apply to a logic satisfying L1 to L3 and a …nite agenda.8
8In Wilson’s model, the notion of consistency (feasibility) rather than that of entailment
is a primitive. This is slightly less general than our model, as the notion of entailment fully speci…es a notion of consistency, while the converse does not hold for all logics satisfying L1 to L3.
Although we have constructed an explicit embedding of preference aggre-gation into judgment aggreaggre-gation, we have not proved the impossibility of a converse embedding. We suspect that such an embedding is hard to achieve, as Arrow’s standard model cannot easily capture the di¤erent informational basis of judgment aggregation. It is unclear what an embedding of judgment aggrega-tion into preference aggregaaggrega-tion would look like. In particular, it is unclear how to specify theoptions over which individuals have preferences. Thepropositions
in an agenda are not candidates for options, as propositions are usually not mu-tually exclusive. Natural candidates for options are perhaps entirejudgment sets
(complete and consistent), as these are mutually exclusive and exhaustive. But in a preference aggregation model with options thus de…ned, individuals would feed into the aggregation rule not a single judgment set (option), but an entire preference ordering over all possible judgment sets (options). This would be a di¤erent informational basis from the one in judgment aggregation. In addition, the explicit logical structure within each judgment set would be lost under this approach, as judgment sets in their entirety, not propositions, would be taken as primitives. However, although we are sceptical, the construction of a useful converse embedding remains an open challenge.
6
References
Bovens L, Rabinowicz W (2004) Democratic Answers to Complex Ques-tions: An Epistemic Perspective. Synthese (forthcoming)
Dietrich F (2004a) Judgment Aggregation: (Im)Possibility Theorems. Jour-nal of Economic Theory (forthcoming)
Dietrich F (2004b) Judgment aggregation in general logics. Working paper, PPM Group, University of Konstanz
Dietrich F, List C (2005) Judgment aggregation by quota rules. Working paper, LSE
Dokow E, Holzman R (2005) Aggregation of binary relations, Working pa-per, Technion Israel Institute of Technology
Gärdenfors P (2004) An Arrow-like theorem for voting with logical conse-quences. Economics and Philosophy (forthcoming)
List C (2003) A Possibility Theorem on Aggregation over Multiple Intercon-nected Propositions. Mathematical Social Sciences 45(1): 1-13
List C (2004) A Model of Path Dependence in Decisions over Multiple Propo-sitions. American Political Science Review 98(3): 495-513
List C (2005) The Probability of Inconsistencies in Complex Collective De-cisions. Social Choice and Welfare 24: 3-32
List C, Pettit P (2002) Aggregating Sets of Judgments: An Impossibility Result. Economics and Philosophy 18: 89-110
List C, Pettit P (2004) Aggregating Sets of Judgments: Two Impossibility Results Compared. Synthese 140(1-2): 207-235
Nehring K (2003) Arrow’s theorem as a corollary. Economics Letters 80: 379-382
Nehring K, Puppe C (2005) Consistent Judgment Aggregation: A Charac-terization. Working paper, University of Karlsruhe
Pauly M, van Hees M (2004) Logical Constraints on Judgment Aggregation. Journal of Philosophical Logic (forthcoming)
Pettit P (2001) Deliberative Democracy and the Discursive Dilemma. Philo-sophical Issues 11: 268-299
Pigozzi G (2004) Collective decision-making without paradoxes: An argument-based account. Working paper, King’s College, London
van Hees M (2004) The limits of epistemic democracy. Working paper, University of Groningen
Wilson R (1975) On the Theory of Aggregation. Journal of Economic The-ory 10: 89-99
A
Appendix
Proof of Proposition 1. Let X be minimally connected and let F be any ag-gregation rule. Put N := f1; :::; ng. If F is dictatorial, F obviously satis-…es universal domain, collective rationality, systematicity and the unanimity principle. Now assume F satis…es the latter conditions. Then there is a set
C of ("winning") coalitions C N such that, for every p 2 X and every
(A1; :::; An) 2 Domain(F), F(A1; :::; An) = fp 2 X : fi : p 2 Aig 2 Cg. For every consistent setZ X;let AZ be some consistent and complete judgment set such that Z AZ.
Claim 1. N 2 C, and, for every coalition C N, C 2 C if and only if
NnC =2 C.
The …rst part of the claim follows from the unanimity principle, and the second part follows from collective rationality together with universal domain.
Claim 2. For any coalitions C; C N; if C 2 C and C C then C 2 C. Let C; C N with C 2 C and C C . Assume for contradiction that
C =2 C. ThenNnC 2 C. LetY be as in part (ii) of the de…nition of minimally connected agendas, and letZ be asmallest subset ofY such that(YnZ)[f:z :
z 2 Zg is consistent and Z has even size. We have Z 6=;; since otherwise the (inconsistent) set Y would equal the (consistent) set (YnZ)[ f:z : z 2 Zg. So, as Z has even size, there are two distinct propositions p; q 2Z. Since Y is minimal inconsistent,(Ynfpg)[ f:pg and (Ynfqg)[ f:qgare each consistent. This and the consistency of (YnZ)[ f:z : z 2 Zg allow us to de…ne a pro…le
(A1; :::; An) 2 Domain(F) as follows. Putting C1 := C nC and C2 := NnC
(note thatfC; C1; C2g is a partition of N), let
Ai := 8 < : A(Ynfpg)[f:pg if i2C A(YnZ)[f:z:z2Zg if i2C1 A(Ynfqg)[f:qg if i2C2. (1)
p q C C1 C2 p q r C1 C2 C 0
Figure 1: The pro…les constructed in the proofs of claims 2 (left) and 3 (right).
By (1), we have YnZ F(A1; :::; An) as N 2 C. Also by (1); we have q 2
F(A1; :::; An) asC 2 C, andp2F(A1; :::; An)asC2 =NnC 2 C. In summary,
writingZ :=Znfp; qg, we have (*) YnZ F(A; :::; An): We distinguish two cases.
Case C1 2 C= . Then C [C2 = NnC1 2 C. So Z F(A1; :::; An) by (1), which together with (*) implies Y F(A1; :::; An). But then F(A1; :::; An) is inconsistent, a contradiction.
Case C1 2 C. So f:z :z 2 Z g F(A1; :::; An) by(1). This together with (*) implies that (YnZ )[ f:z : z 2 Z g F(A1; :::; An). So (YnZ )[ f:z :
z2Z gis consistent. AsZ also has even size, the minimality condition in the de…nition of Z is violated.
Claim 3. For any coalitions C; C N; if C; C 2 C then C\C 2 C. Consider anyC; C 2 C. LetY Xbe as in part (i) of the de…nition of min-imally connected agendas. As jYj 3, there are pairwise distinct propositions
p; q; r 2 Y. As Y is minimally inconsistent, each of the sets (Ynfpg)[ f:pg,
(Ynfqg)[ f:qg and (Ynfrg)[ f:rg is consistent. This allows us to de…ned a pro…le(A1; :::; An)2Domain(F)as follows. PuttingC0 :=C\C ,C1 :=C nC
and C2 :=NnC (note thatfC0; C1; C2g is a partition ofN), let
Ai := 8 < : A(Ynfpg)[f:pg if i2C0 A(Ynfrg)[f:rg if i2C1 A(Ynfqg)[f:qg if i2C2. (2) By (2); Ynfp; q; rg F(A1; :::; An) as N 2 C. Again by (2), we have q 2 F(A1; :::; An)asC0[C1 =C 2 C. AsC 2 CandC C0[C2, we haveC0[C2 2
Cby claim 2. So, by(2); r 2F(A1; :::; An). In summary,Ynfpg F(A1; :::; An): As Y is inconsistent, p =2 F(A1; :::; An); and hence :p 2 F(A1; :::; An): So, by
(2), C0 2 C.
Claim 4. There is a dictator.
Consider the intersection of all winning coalitions, Ce := \C2CC: By claim 3,Ce 2 C. So Ce6=;, as by claim 1;2 C= . Hence there is aj 2C:e As j belongs to every winning coalition C 2 C, j is a dictator: indeed, for each pro…le
(A1; :::; An)2Domain(F) and each p2X, if p2Aj then fi:p 2Aig 2 C, so that p2 F(A1; :::; An); and if p =2 Ai then :p2 Ai, so that fi : :p2 Aig 2 C, implying:p2F(A1; :::; An), and hencep =2F(A1; :::; An).
Proof of Theorem 1. Let X be minimally connected, and let F satisfy universal domain, collective rationality and systematicity. If F satis…es the unanimity principle, then, by Proposition 1, F is dictatorial. Now suppose F
violates the unanimity principle.
Claim 1. X is symmetrical, i.e. if A X is consistent, so is f:p:p2Ag. Let A X be consistent. Then there exists a complete and consistent judgment set B such that A B. As F violates the unanimity principle (but satis…es systematicity), the setF(B; :::; B)contains no element ofB, hence con-tains no element ofA, hence contains all elements off:p:p2Agby collective rationality. So, again by collective rationality,f:p:p2Ag is consistent.
Claim 2. The aggregation rule Fb with universal domain de…ned by b
F(A1; :::; An) := f:p:p2F(A1; :::; An)g is dictatorial.
As F satis…es collective rationality and systematicity, so does Fb, where the consistency of collective judgment sets follows from claim 1. Fb also satis…es the unanimity principle: for any p 2 X and any (A1; :::; An) in the universal domain, where p 2 Ai for all i, p =2 F(A1; :::; An), hence :p 2 F(A1; :::; An), and sop=::p2Fb(A1; :::; An). Now Proposition 1 applies to Fb, and henceFb is dictatorial.
Claim 3. F is inverse dictatorial.
The dictator forFb is an inverse dictator forF.
Proof of Lemma 1. LetX and F be as speci…ed. To show that F is system-atic, consider anyp; q 2X and any(A1; :::; An);(A1; :::; An)2Domain(F)such thatC :=fi:p2Aig=fi:q2Aig, and let us prove that p2F(A1; :::; An)if and only ifq2F(A1; :::; An). Ifpandqare both tautologies (f:pgandf:qgare inconsistent), the latter holds since (by collective rationality)p2F(A1; :::; An) and q 2 F(A1; :::; An). If p and q are both contradictions (fpg and fqg are inconsistent), it holds since (by collective rationality) p =2 F(A1; :::; An) and
q =2 F(A1; :::; An). It is impossible that one of p and q is a tautology and the other a contradiction, because then one of fi:p 2Aig and fi:q 2 Aig would beN and the other;.
Now consider the remaining case where both p and q are contingent. We say that C is winning for r (2 X) if r 2 F(B1; :::; Bn) for some (hence by independence any) pro…le (B1; ::; Bn) 2 Domain(F): with fi : r 2 Big = C. We have to show that C is winning for p if and only if C is winning for q. Suppose C is winning for p, and let us show that C is winning for q (the converse implication can be shown analogously). As X is path-connected and
p and q are contingent, there are p=p1; p2; :::; pk =q 2 X such that p1 p2,
p2 p3, ..., pk 1 pk. We show by induction that C is winning for each of
and assume C is winning for pj. We show that C is winning for pj+1. By
pj pj+1, there is a set Y X such that (i) fpjg [Y and f:pj+1g [Y are
each consistent, and (ii) fpj;:pj+1g [Y is inconsistent. Using (i) and (ii), the
setsfpj; pj+1g [Y and f:pj;:pj+1g [Y are each consistent. So there exists a
pro…le(B1; :::; Bn)2Domain(F)such thatfpj; pj+1g[Y Bi for alli2Cand
f:pj;:pj+1g [Y Bi for alli =2C. Since Y Ai for all i, Y F(A1; :::; An) by the unanimity principle. Since fi : pj 2 Aig = C is winning for pj, we have pj 2 F(A1; :::; An). So fpjg [Y F(A1; :::; An). Hence, using collective rationality and (ii), we have:pj+12=F(A1; :::; An), and sopj+1 2F(A1; :::; An). Hence, asfi:pj+1 2Aig=C, C is winning forpj+1.
Proof of Lemma 2. Let X be the preference agenda. X is minimally con-nected, as, for any pairwise distinct constants x; y; z, the set Y =
fxP y; yP z; zP xg X is minimal inconsistent, where f:xP y;:yP z; zP xg is consistent.
To prove path-connectedness, note that, by the axioms of our predicate logic for representing preferences, (*) :xP y and yP x are equivalent (i.e. entail each other) for any distinctx; y 2K. Now consider any (contingent)p; q 2X, and let us construct a sequence p = p1; p2; :::; pk =q 2 X with p j= p2; :::; pk 1 j= q.
By (*), if p is a negated proposition :xP y, then p is equivalent to the non-negated propositionyP x ; and similarly for q. So we may assume without loss of generality that p and q are non-negated propositions, say p is xP y and q is
x0P y0. We distinguish three cases, each with subcases.
Case x= x0. If y =y0, then xP y xP y =x0P y0 (take Y =;). If y 6=y0, then xP y xP y0 =x0P y0 (take Y =fyP y0g).
Case x=y0. Ify=x0, then, taking anyz 2Knfx; yg, we have xP y xP z (take Y = fyP zg), xP z yP z (take yP x), and yP z yP x = x0P y0 (take
Y = fzP xg). If y 6= x0, then xP y x0P y (take Y = fx0P xg) and x0P y x0P y0 (take Y =fyP y0g).
Case x 6= x0; y0. If y = x0, then xP y xP y0 (take Y = fyP y0g) and
xP y0 x0P y0 (take Y =fx0P xg). If y =y0, then xP y x0P y = x0P y0 (take
